1\DOC EXISTS_AND_CONV
2
3\TYPE {EXISTS_AND_CONV : conv}
4
5\SYNOPSIS
6Moves an existential quantification inwards through a conjunction.
7
8\KEYWORDS
9conversion, quantifier, existential, conjunction.
10
11\DESCRIBE
12When applied to a term of the form {?x. P /\ Q}, where {x} is not free in both
13{P} and {Q}, {EXISTS_AND_CONV} returns a theorem of one of three forms,
14depending on occurrences of the variable {x} in {P} and {Q}.  If {x} is free
15in {P} but not in {Q}, then the theorem:
16{
17   |- (?x. P /\ Q) = (?x.P) /\ Q
18}
19is returned.  If {x} is free in {Q} but not in {P}, then the
20result is:
21{
22   |- (?x. P /\ Q) = P /\ (?x.Q)
23}
24And if {x} is free in neither {P} nor {Q}, then the result is:
25{
26   |- (?x. P /\ Q) = (?x.P) /\ (?x.Q)
27}
28
29
30\FAILURE
31{EXISTS_AND_CONV} fails if it is applied to a term not of the form
32{?x. P /\ Q}, or if it is applied to a term {?x. P /\ Q} in which the
33variable {x} is free in both {P} and {Q}.
34
35\SEEALSO
36Conv.AND_EXISTS_CONV, Conv.EXISTS_AND_REORDER_CONV, Conv.LEFT_AND_EXISTS_CONV, Conv.RIGHT_AND_EXISTS_CONV.
37\ENDDOC
38